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  "body": "\n## Summary\n\nA Maass wave form is a smooth function on the [[upper half-plane|X6XGGH5FPW]] that is invariant under a [[congruence subgroup|1H6R856P3C]] of the [[modular group|VEKV1EM77B]].\nIt is also an eigenfunction of the hyperbolic [[Laplace–Beltrami operator|F4NWZNTRMD]].\nIt is the non-holomorphic counterpart of a [[modular form|GWKN5VG78A]].\nMaass forms are basic examples of [[automorphic forms|3C73HND6XA]], and their eigenvalues are the subject of spectral theory and trace formulas.\n\n## Explanations\n\n### L1 human\n\nThink of the upper half-plane as a drum whose shape is folded up by the modular group.\nA Maass form is a standing wave on this drum: a smooth wave that repeats under the symmetries and has a pure \"frequency\", its Laplace eigenvalue.\nHolomorphic modular forms are rigid objects built from complex analysis; Maass forms are the wave-like analogues that need no complex differentiability.\nA simple example is the real-analytic Eisenstein series $E(z, s)$, an average of the function $y^s$ over the modular group.\nSquare-integrable Maass forms for $\\mathrm{SL}_2(\\Z)$ exist, but no one knows a closed formula for any of them.\n",
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